Question 1

Sketch the region R and evaluate the iterated integral.  &#10;A)    &#10;B)    &#10;C)    &#10;D)    &#10;E)

Accepted Answer

11eaa595_62c1_a787_a696_f1a964539f86_TB4584_11_TB4584_11 11eaa595_62c1_ce98_a696_c5fcf970a73b_TB4584_11_TB4584_11

Question 2

Use an iterated integral to find the area of the region bounded by   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E) The integral is improper and does not converge.

Accepted Answer

11eaa595_62be_24f7_a696_29659dbc9def_TB4584_11_TB4584_11

Question 3

Evaluate the following integral.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

11eaa595_62b8_f4a9_a696_0978be6ca562_TB4584_11_TB4584_11

Question 4

Evaluate the following iterated integral.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Evaluate the following iterated integral.  &#10;A)...

Question 5

Evaluate the following iterated integral.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Evaluate the following iterated integral.  &#10;A)...

Question 6

Evaluate the improper iterated integral   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

Using Fubini's theorem, we can switch the order of integration and evaluate the integral as follows:
 
∫[0,1/2] ∫[2y,1] e^(xy) dx dy

This is a simple integration problem with the result:

(1/2) * (e^(y(2-y)) - 1)

Integrating  this from 0 to 2 gives:

(1/2) * (e^4 - 1 - e^0 + 1) = (e^4 - 1)/2

Therefore, the correct answer is D.

Question 7

Use an iterated integral to find the area of the region shown in the figure below.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

We can integrate with respect to $y$ first since the bounds of $y$ are constants. The range of $y$ is from $0$ to $4$. For a fixed value of $y$, we need to find the horizontal limits of integration, which are given by the curves $y=x^2-2$ and $y=x-2$, or $x=\sqrt{y+2}$ and $x=y+2$, respectively. Thus, the area is given by \begin{align*}
\int_0^4 \int_{\sqrt{y+2}}^{y+2} dx dy &= \int_0^4 \left( y+2 - \sqrt{y+2}ight) dy\
&= \left[\frac{1}{2}y^2 + 2y - \frac{2}{3}(y+2)^{3/2}ight]_0^4\
&= \boxed{2\sqrt{2}}.
\end{align*}

Question 8

Evaluate the iterated integral   by switching the order of integration. Round your answer to three decimal places.&#10;A) 0.538&#10;B) 0.558&#10;C) 30.538&#10;D) 1.538&#10;E) 4.538

Accepted Answer

The answer of Evaluate the iterated integral   by...

Question 9

The area of a region R is given by the iterated integral   . Switch the order of integration and show that both orders yield the same area. What is this area?&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

First, let's write out the boundaries of integration for both orders:

Original:  $R=\int_{-1}^1\int_x^{\sqrt{1-x^2}} e^{y^2}\,dy\,dx$
Reverse:  $R=\int_{-1}^1\int_{-\sqrt{1-y^2}}^y e^{y^2}\,dx\,dy$

To switch the order of integration, we need to imagine the region $R$ as being bounded by the lines $x=a$ (the left boundary of the region) and $y=b$ (the top boundary of the region) where $a$ and $b$ vary over the same ranges as in the original order of integration. Then we integrate with respect to $a$ from $-1$ to $b$ and with respect to $b$ from $-1$ to 1. This gives:

$R=\int_{-1}^1\int_{a}^{\sqrt{1-a^2}} e^{y^2}\,dy\,da$

Notice that this is almost identical to the original order of integration, except that we now integrate $a$ from $-1$ to $b$ instead of $x$ from $-1$ to $1$. To finish, we need to change the order of integration in this expression as well:

$R=\int_{-1}^1\int_{-1}^{y} e^{y^2}\,da\,dy$

This is the same as the reverse order of integration above, since $a$ ranges from $-1$ to $y$ in both cases. Therefore, we have shown that switching the order of integration does not change the value of the integral. Evaluating this integral gives:

$R=\int_{-1}^1\int_{x}^{\sqrt{1-x^2}} e^{y^2}\,dy\,dx=\int_{-1}^1\int_{-\sqrt{1-y^2}}^y e^{y^2}\,dx\,dy\approx2.25$

The correct answer is (C).

Question 10

Evaluate the following improper integral.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E) The integral does not converge.

Accepted Answer

The answer of Evaluate the following improper integral.  &#10;A)...

Question 11

Evaluate the following integral.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Evaluate the following integral.  &#10;A) ...

Question 12

Evaluate the following iterated integral.  &#10;A) 5,901&#10;B) 5,931&#10;C) 5,955&#10;D) 5,970&#10;E) 6,017

Accepted Answer

The answer of Evaluate the following iterated integral.  &#10;A)...

Question 13

Sketch the region R of integration and then switch the order of integration for the following integral.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Sketch the region R of integration and...

Question 14

Evaluate the following iterated integral.  &#10;A) 41.5&#10;B) 38.5&#10;C) 34.5&#10;D) 33.5&#10;E) 37.5

Accepted Answer

The answer of Evaluate the following iterated integral.  &#10;A)...

Question 15

Evaluate the iterated integral below. Note that it is necessary to switch the order of integration.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Evaluate the iterated integral below. Note that...

Question 16

Use an iterated integral to find the area of the region bounded by   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Use an iterated integral to find the...

Question 17

The area of a region R is given by the iterated integral   . Switch the order of integration and show that both orders yield the same area. What is this area?&#10;A) 65&#10;B) 16&#10;C) 21&#10;D) 8&#10;E) 4

Accepted Answer

The answer of The area of a region R is...

Question 18

Use an iterated integral to find the area of the region bounded by the graphs of the equations   and   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Use an iterated integral to find the...

Question 19

The area of a region R is given by the iterated integrals   . Switch the order of integration and show that both orders yield the same area. What is this area?&#10;A) 32&#10;B) 1025&#10;C) 64&#10;D) 771&#10;E) 63

Accepted Answer

The answer of The area of a region R is...

Question 20

Evaluate the following integral.  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Evaluate the following integral.  &#10;A) ...

Deck 14: Section 1: Multiple Integration